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V Relation with general scaling theory

[] The results of the previous section are closely related with the general classification theory of phase transitions [18, 19, 20, 21, 22, 23], the probabilistic approach in the theory of critical phenomena [34, 35, 36], and finite size scaling theory for the order parameter distribution [7]. [] The relation with the general classification theory of phase transitions [21, 22] has already been given above. [] The relation with the probabilistic approach to critical phenomena [35] is that scaling and universality are obtained probabilistically from stability and nonempty domains of attraction for stable distributions. [] The difference to [34, 35, 36] is that in those works the usual scaling limit of the measure [page 72, §0]    \mu(\mbox{\boldmath$\vec{\varphi}$\unboldmath},a,L,\mbox{\boldmath$\vec{\Pi}$\unboldmath}) is studied instead of the much simpler distributions appearing in the ensemble limit. [] Similarly the differences with finite size scaling theory of the order parameter distribution [7] arise from the difference between the finite size scaling limit and the ensemble limit.

[] The relation with the renormalization group scaling theory of critical points [37] is provided by the identification (4.15) relating the thermodynamic fluctuation exponents to the field theoretic correlation exponents, i.e. by hyperscaling. [] The present theory considers only relevant operators by virtue of the general inequality \varpi _{X}>0. [] Note that marginal operators correspond formally to \varpi _{X}\rightarrow\pm\infty, not to \varpi _{X}\rightarrow 0. [] The influence of irrelevant operators is reflected in the general presence of a slowly varying function \Lambda(x) in all scaling relations.

[] The traditional classification into irrelevant (IO), marginal (MO) and relevant operators (RO) can be extended by three additional distinctions. [] The first refinement is into equilibrium (ERO) and anequilibrium relevant operators (ARO) according to y_{{ERO}}\leq d for equilibrium relevant operators and y_{{ARO}}>d for anequilibrium relevant operators. [] ARO’s are readily constructed from ERO’s and are well known to occur in many models. [] Examples are non-primary operators in conformal field theory [17], the energy and order parameter in anequilibrium phase transitions [21, 22], high gradient operators in the O(n) nonlinear \sigma models [38, 39] or the hierarchical shell number modes in shell models for turbulence [40]. [] An intriguing formal analogy exists between the random local events building up a multifractal measure and anequilibrium relevant operators [41].

[] A second refinement of the traditional classification is to distinguish between gaussian and nongaussian relevant operators. [] A relevant operator X is called gaussian if y_{X}\leq d/2 and nongaussian if y_{X}>d/2. [] By virtue of the duality law [28]

h(x;\varpi _{X},\zeta _{X},0,1)=x^{{-1-\varpi _{X}}}h(x^{{-\varpi _{X}}};1/\varpi _{X},\zeta _{X}^{\prime},0,1,) (5.1)

where \zeta _{X}^{\prime}=\zeta _{X}^{\prime}(\varpi _{X},\zeta _{X}) an additional third distinction is expected for operators with y_{X}<2d as compared to those with y_{X}\geq 2d. [] The precise nature of this distinction remains to be explored.

[] The new extended classification of the spectrum of critical operators may (in obvious notation) be summarized by the inequalities

y_{{IO}}<0=y_{{MO}}<y_{{GERO}}\leq d/2<y_{{NERO}}\leq d<y_{{ARO_{1}}}\leq 2d<y_{{ARO_{2}}} (5.2)

in which the relevance increases from left to right.